I think that I have done enough
to show that it is quite possible to understand how a convergent-divergent
nozzle “works” and I have shown that useful calculations can be made to the
point where an engineer can make a good start at designing a nozzle. However it
makes sound sense to look at the nozzle in use to see what constraints the
engineering imposes on the physics.

I can think of only three uses
for convergent-divergent nozzles. There must be more but these three will serve
my purpose. They are the nozzles in a steam turbine, the nozzles of a rocket
engine and the intake duct of a gas turbine engine for supersonic flight. Let
me look at these in turn.

Steam turbines used for electricity
generation can use steam that is supplied at up to 170 bar[1]
(2,500 psi) and they all exhaust at something like 0.1 bar. Of course there are
other smaller turbines working with steam at lower pressures. The field of
application is very extensive and the study of these turbines is a vast
undertaking. Nevertheless there are a few basic principles that can prove to be
useful. A range of pressure from 170 bar to 0.1 bar is an extreme range and
graph 13-15 shows more or less how the pressure varies with volume during the
expansion in the steam turbine. The possible area under this curve is very
small indeed for the ranges of pressure and volume involved. Only a steam
turbine can make use of this range to extract work from the steam.

Steam turbines do not make one
continuous expansion, they often make the expansion in three or four stages.
This gives two engineering advantages, the first is to give opportunities for
improvements in efficiency by heating the steam between stages and the second
is that the thickness of the casings can be matched to the pressures that are
exerted on them.

It is not unreasonable to think
in terms of having each stage produce the same power. The power produced in a
stage of a turbine is dependent on and in graph 13-16 I limited the ranges of
pressure and volume so that areas can actually be seen on the graph. Then I
have roughly divided the total area under an expansion into four equal areas.
It immediately becomes evident that the first stage will involve a large
pressure drop and small volumes of steam and for the last stage will be at low
pressure and with very large volumes of steam.

We have seen that sonic speeds
are associated with pressure ratios of 50+% and the pressure ratio on the first
stage have about this value. As a result steam turbines often use a de Laval
nozzle to extract work from the steam as kinetic energy and then transfer this
kinetic energy to one or more rows of rotating blades and then the steam is
expanded continuously in many stages each consisting of a fixed row of blades
followed by a row of moving blades through the remaining three stages. In the
first stage the blades operate at constant pressure and the best way to use the
steam is to make a big drop in pressure in a ring of de Laval nozzles to start
the expansion and have the blades operate in a drum at a much lower pressure
than the boiler pressure. This is called an impulse stage and we now have the
expressions that might let us design a nozzle.

Now
we need to see what the mechanics of an impulse stage look like. I think that
it is important to draw them reasonably accurately and not hide behind the
words “schematic” or “ not to scale”. Figure 13-10 shows how two cambered,
aerofoil-shaped blades can form a convergent-divergent nozzle or, indeed, just
a convergent nozzle. These blades can be robust. The moving blades must also be
robust just to stand the forces of very high-speed steam of high-density
flowing over them and the centrifugal forces when rotating at 3,000 rpm as
well. Fortunately they need to have this shape anyway to give the best flow
pattern.

Look at the space between two
fixed blades. It is a convergent-divergent nozzle but it is cropped off
obliquely. This is done to ensure that the jet of steam has no unguided
distance in which to start to mix and so lose kinetic energy to random motion
and the gap between the exit from the fixed row and entry to the moving row is
kept to a minimum. There is a price to be paid for cropping the nozzle in this
way that is that the jet is diverted slightly as it flows with guidance on only
one side but this is preferable to having a gap.

The nozzles are created in an
annular ring and this ring has to withstand the very high pressure. It is not
easy to ensure the surface finish of the nozzles. The engineering constraints
of this design limit the value of attempts to refine the physics of the
convergent-divergent nozzle as I have laid it out in this text. Nevertheless
practical nozzles have efficiencies of 95%
and higher.

Rockets are very simple engines. I have
attempted to draw the essential features in figure 13-11. The design is
dominated by the need to minimise its weight. Fuel and oxidant are pumped at
very high rates into the combustion chamber where they combine to produce an
enormous flow of gas. The creation of this gas and the resistance to its escape
through the nozzle produces a high pressure in the combustion chamber. The gas
flows out through the convergent-divergent nozzle under the pressure difference
between the gas in the combustion chamber and the outside conditions. I have
also drawn a series of arrows to indicate the way in which pressure is exerted
on the inner surfaces much the same way as is commonly used to show pressure
distribution over an aerofoil. All over the outer surface the pressure is
whatever it may be around the engine. The pressure across the exit plane is not
so easily known and I may have to deal with it separately.

I explained how these pressure
distributions give rise to a net force in chapter 5 in the text associated with
figures 5-14 and 5-15a and b. In essence there is a very large upward force on
the top of the combustion chamber that is not balanced by a downward force on
the convergent part of the nozzle so that there is a net upward force on the
combustion chamber exerted by the gas. This force is exerted on the engine
mounting and then on to whatever is being driven.

In the divergence the pressure
falls steadily but, throughout, this pressure exerts a distributed force on the
bell-shaped divergence that everywhere has a vertical component and there is a
net upward force produced on the divergence. That force is exerted through the
combustion chamber on to the engine mounting together with that from the
combustion chamber.

This is simple in principle but
not so easy to produce as an efficient mechanical device. I presume that it
would be desirable to complete the combustion in the combustion chamber and,
for this to be possible, the combustion chamber would have to be quite large. As
it is a pressure vessel, if it is to be large it would also be heavy and there
is a great incentive to keep its size to a minimum. If the combustion chamber
were to be too small the combustion would take place partially in the nozzle. I
have said that I cannot tell the difference between a hot luminous gas and a
burning gas so just looking at pictures of rockets does not help. A compromise
must be made between weight and performance and that compromise is affected by
other factors.

Photographs of the inner surfaces
of rocket nozzles suggest that they are lined with refractory material and it
is not really smooth and, even if it were to be, it is not likely to survive a
protracted burn unscathed.

The booster rocket works in just
the same way as a liquid fuelled engine but now the upward force on the casing
containing the burning fuel is exerted either on the fuel itself if it is not
porous to gas or on the inner surface of the top of the casing if it is porous.
So instead of exerting a force on the bottom of the rocket as the
liquid-fuelled one does the force probably acts at least in part on the top.
Fuel cases for rockets are columns and are designed as such[2].

So whilst we have modelled the
convergent-divergent nozzle as an adiabatic frictionless flow the reality is
very different. Nevertheless the model
is an enormous help in getting a basic understanding of rocket nozzles and of
possible ways to store experimental data.

The supersonic gas turbine engine is now used only in military
applications. The gas turbine engine cannot operate with supersonic flow
anywhere over its blading but it is used to power supersonic aeroplanes. The
inlet conditions for the Olympus engines used on the Concorde require
inlet conditions that are effectively those of ambient conditions at sea level.
This may not be accidental because those engines, running on gas, also power
electricity generators.

An aeroplane capable of
supersonic speeds must also fly subsonically over a range of speeds up to 1,000
km/h. Then it must go on to fly at perhaps 3,000 km/h. There is only one way to
cope with this and that is to fit intake ducts that can be used in conjunction
with an engine management system to change the shape of the duct from one
suitable for low speed to one suitable for high speed. At low speed it is
essentially a straight through duct but at high speed the duct is a
convergent-divergent nozzle with air entering it supersonic speed, possibly at
low pressure and low temperature, and leaving the duct to enter the engine at about
1 bar absolute and about 300°K.
The duct is rectangular in cross-section as the only possible design. In the
supersonic configuration the air slows down in the convergence and slows down
in the divergence with a steady increase in pressure. Surprisingly the net
effect is that the ducts produce a very substantial forwards force that
contributes about 40% of the total thrust. This sounds odd but we seldom ask
the question “ how is the force produced by a jet engine and where is it
exerted on the frame of the engine?” There are surprising answers.

[1] This unlikely looking figure is almost the
maximum pressure that that is practical for steam plant.